Note on beta elements in homotopy, and an application to the prime three case

Author:
Katsumi Shimomura

Journal:
Proc. Amer. Math. Soc. **138** (2010), 1495-1499

MSC (2010):
Primary 55Q45

DOI:
https://doi.org/10.1090/S0002-9939-09-10190-9

Published electronically:
December 8, 2009

MathSciNet review:
2578544

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Abstract: Let denote the sphere spectrum localized at an odd prime . Then we have the first beta element , whose cofiber we denote by . We also consider a generalized Smith-Toda spectrum characterized by . In this note, we show that an element of gives rise to a beta element of homotopy groups of spheres. As an application, we show the existence of at the prime three to complete a conjecture of Ravenel's: exists if and only if is not congruent to , or mod .

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Additional Information

**Katsumi Shimomura**

Affiliation:
Department of Mathematics, Faculty of Science, Kochi University, Kochi, 780-8520, Japan

Email:
katsumi@kochi-u.ac.jp

DOI:
https://doi.org/10.1090/S0002-9939-09-10190-9

Received by editor(s):
April 19, 2009

Published electronically:
December 8, 2009

Communicated by:
Brooke Shipley

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.